bioRxiv preprint first posted online Aug. 13, 2015; doi: http://dx.doi.org/10.1101/024562. The copyright holder for this preprint (which was
not peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
The role of oxygen in avascular tumor growth
David Robert Grimes1 , Pavitra Kannan1 , Alan McIntyre2 , Anthony Kavanagh3 , Abul
Siddiky1 , Simon Wigfield2 , Adrian Harris2 , Mike Partridge1
1 Cancer Research UK/MRC Oxford Institute for Radiation Oncology, Gray
Laboratories, University of Oxford, Old Road Campus, Oxford, OX3 7DQ
2 The Weatherall Institute for Molecular Medicine, University of Oxford,
John Radcliffe Hospital/Headley Way, Oxford, OX3 9DS
3 Advanced Technology Development Group, Department of Oncology,
University of Oxford, Old Road Campus Research Building, Oxford, OX3
7DQ
* [email protected]
Abstract
The oxygen status of a tumor has significant clinical implications for treatment prognosis,
with well-oxygenated subvolumes responding markedly better to radiotherapy than poorly
supplied regions. Oxygen is essential for tumor growth, yet estimation of local oxygen
distribution can be difficult to ascertain in situ, due to chaotic patterns of vasculature. It
is possible to avoid this confounding influence by using avascular tumor models, such as
tumor spheroids, where oxygen supply can be described by diffusion alone and are a much
better approximation of realistic tumor dynamics than monolayers. Similar to in situ
tumours, spheroids exhibit an approximately sigmoidal growth curve, often approximated
and fitted by logistic and Gompertzian sigmoid functions. These describe the basic
rate of growth well, but do not offer an explicitly mechanistic explanation. This work
examines the oxygen dynamics of spheroids and demonstrates that this growth can be
derived mechanistically with cellular doubling time and oxygen consumption rate (OCR)
being key parameters. The model is fitted to growth curves for a range of cell lines and
derived values of OCR are validated using clinical measurement. Finally, we illustrate
how changes in OCR due to gemcitabine treatment can be directly inferred using this
model.
1
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Author Summary
We derive a mechanistic growth model for avascular tumors, yielding a familiar sigmoidal
growth curve with a minimum of assumptions. Specifically, it is assumed that only cells
with requisite oxygen for mitosis can produce daughter cells. This work is validated
on tumor spheroids, with well-understood oxygen dynamics and distributions and found
to fit the measured data from several cell lines well. The importance of cellular mass in
determining OCR is outlined, and a method for doing so alongside theoretical justification
is outlined. Finally, the application of the model in determining the change in OCR due
to clinical compounds is demonstrated using gemcitabine, a potent radio-sensitizer.
Introduction
Tumor spheroids are clusters of cancer cells which grow in approximately spherical 3D
aggregates. This property makes them a useful experimental model for avascular tumor growth. Spheroids are preferred over 2D monolayers in several applications as the
signalling and metabolic profiles are more similar to in vivo cells than standard monolayers [1]. Like monolayers, spheroids are relatively straightforward to culture and examine.
For these reasons, spheroids have been widely used to investigate the development and
consequences of tissue hypoxia. [1]. Early investigations using spheroids began in earnest
in the 1970s [2], and the nature of spheroid growth has long been an active question,
with several interesting properties mimicing solid tumors. Conger & Ziskin [3] analysed
the growth properties of tumor spheroids and noted that they appeared to grow in three
distinct stages; exponentially, approximately linearly and then reaching a plateau. A
similar type of growth was seen over 15 different tumor cell lines [4], and it was observed
that this growth could be approximated to a Gompertzian curve, which described the
approximate sigmoidal shape of the growth curves well. In recent years, there has been
renewed interest in tumor spheroids in general and the scope for their application has
increased dramatically - spheroids have been used in radiation biology [5–8] as a means to
test fractionation and other parameters in a controllable environment, in chemotherapy
to act as a model for drug delivery [9–12] and even to investigate cancer stem cells [13].
Cancer spheroids have also shown potential as a model for exploring FDG-PET dynamics [14] to explore hypoxia effects in solid tumors.
The distinct sigmoidal growth curves seen in spheroids also occur in some solid tumors,
prompting investigation into whether any appropriate sigmoidal curve could be tempered
to describe spheroid growth, including the von Bertalanffy and logistic family of models.
2
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It has been shown by Feller as early as the 1940s [15] that statistical inference alone
could not discriminate between such models; while initially it was postulated that any
sigmoid shape may be adequate [16], later analysis [17] found that while the sigmoid
shape was a pre-requisite to describe spheroid growth, it is not a solely sufficient condition. Gompertzian models have also been used, and have the advantage of being well
suited to situations where empirical models are required, such as the optimization of
radiotherapy [17–19]. A hybrid ”Gomp-ex” model [20] was also found to fit observed
spheroid growth curves well [17]; in this model, initial growth is exponential, followed by
a Gompertzian phase when the increasing cell volume reduces the availability of nutrients
to tumor cells. While Gompertzian models of growth can describe the growth of tumor
spheroids well, they are do not directly address the underlying mechanistic or biophysical processes. Several complex models of avascular growth have arisen from the field
of applied mathematics; a review by Roose et al [21] offers an overview of mathematical approaches to modelling avascular tissue, broadly separating published approaching
into either continuum mathematical models employing spatial averaging or discrete cellular automata-type computational models. Continuum models are typically intended to
model in situ tumors before the onset of angiogensis, and tend to have terms for numerous physical phenomena including acidity and metabolic pathways. Other authors have
posited temporal switching of heterogeneous cell types in 2D models in response to a
generic growth factor [22], or travelling wave solutions for a model switching between living and dead cell types [23] and even models for the stress cells experience in an avasuclar
tumor [24]. These models include terms for a wide array of intercellular processes with
varying levels of mathematical elegance and sophistication, but the presence of a large
number of free parameters make direct validation of such models difficult and the models
are not always useful or suitable for in vitro data. Despite extensive investigation from
several avenues, this is still an active problem - a recent review in Cancer Research [25]
stated that new models and analysis are vital if we are to understand the processes in
tumor growth.
In this investigation, we confine our investigation to a simple case to allow us reduce
the number of parameters. Specifically, we shall model the effects of oxygen on spheroid
growth whilst controlling for other potentially confounding factors. Spheroids provide
insight into how avascular tumors propagate; as spheroids increase in size, their central
core becomes anoxic and leads to the formation of two distinct zones - a necrotic core
and a viable rim, as depicted in figure 1. We have recently derived an explicit analytical model for oxygen distribution in spheroids, which accurately predicts properties such
as the extent of the anoxic, hypoxic and viable regions and allows determination of the
3
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oxygen consumption rate from first principles for a spheroid at a given time point [26].
It can further be shown from this analysis that the oxygen consumption rate (OCR) of
a spheroid drives both its oxygen distribution and the physical extent of the anoxic core
rn . In this work, we derive a time-dependent discrete growth model for tumor spheroids,
linking their relative rates of oxygen consumption to their growth curves. The model used
in this work has only has free parameters of oxygen diffusion and oxygen consumption
rate, and average cellular doubling time so that local oxygen partial pressure determines
cell behaviour. We do not model other nutrients such as glucose, which are assumed
to be present in excess in the media. The model is limited to spheroids from a single
cell line. This model can be directly contrasted to experimental data and is validated
across a range of cell lines. We further show how this method might be used to infer the
effect of different clinical compounds on oxygen consumption rate, using gemcitabine as
an illustration.
Methods
Oxygen diffusion & derivation of mechanistic growth model
For a tumor spheroid in a medium with external partial pressure po , oxygen will diffuse
isotropically at a rate D throughout the spheroid whilst being consumed by the tumor
cells. The rate of oxygen consumption, a, has been shown to determine the oxygen tension
throughout the spheroid and the resultant boundaries of different spheroid regions [26].
For a spheroid consuming oxygen at a rate a, the diffusion length rl is the maximum
radius a spheroid can obtain when the partial pressure at the centre (r = 0) is exactly
zero. This corresponds to a spheroid with no central anoxia. It can be shown rl is given by
rl =
r
6Dpo
aΩ
(1)
where Ω = 3.0318 × 107 mmHg kg m−3 is a constant arising from Henry’s law. Oxygen
consumption rate (OCR) is usually expressed as volume of oxygen consumed per unit
mass per unit time. This can be readily subsumed with the Henry’s law constant to yield
aΩ, the OCR in units of oxygen pressure per second. For a spheroid with a radius ro > rl ,
a central anoxic core of radius rn exists, as illustrated in Fig. 1.
The anoxic core, rn , is related to oxygen consumption rate a and spheroid radius ro by
4
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Figure 1: Cross-section of a tumor spheroid of radius ro . The anoxic radius is
denoted by rn . The radius rp depicts the radial extent of pm , the minimal oxygen level
required for mitosis. The orange part of the image is the region rp ≤ r ≤ ro , the purple
part corresponds to rn ≤ r ≤ rp and the central anoxic core (r ≤ rn ) is shown in gray.


1
rn = ro  − cos 
2
arccos 1 −
3
2rl2
ro2
− 2π

 .
(2)
The same analysis [26] allows determination of the partial pressure at any point in the
spheroid. If the minimum oxygen tension required for mitosis is pm where 0 ≤ pm ≤ po ,
then the radius at which this minimum tension is achieved, rp is given by
5
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rp = 2
r

 s
3
2
φ(pm )
1
3rn

cos  arccos −
3
3
φ(pm )

(3)
where φ(pm ) is given by
φ(p) = ro2 +
2rn3 6D(pm − po )
+
ro
Ωa
(4)
These equations describe the oxygen distribution and physical regions of any given
spheroid, but do not state anything about spheroid growth. However, this can be readily
extended. Initially we consider the volume of living, viable cells. This is simply the
difference in volume between spheres of radius ro and rn , expressed as
Va =
4π 3
ro − rn3 .
3
(5)
We further assume that cells below a minimal oxygen threshold pm are unable to undergo
mitosis and that rp is the radius at which p = pm , and rn ≤ rp ≤ ro . The volume of cells
able to proliferate is therefore given by
Vp =
4π 3
ro − rp3 .
3
(6)
If the average cellular doubling time is td , then after this time interval then total cells
after doubling is simply the sum of Va and Vp . This will in turn lead to not only a new
volume and radius roN , but will yield a new anoxic radius rnN and viable radius rpN ,
all of which can be calculated from the oxygen consumption rate. We can express this
model as a piecewise iterative approach modelling oxygen growth, recalculating roN , rnN
and rpN at intervals of the cellular doubling time td . The total volume and radius at the
iteration N + 1 is thus given by
6
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4π
2ro3N − rp3N − rn3 N
3
1/3
= 2ro3N − rp3N − rn3 N
.
VN +1 =
(7)
roN+1
(8)
This growth model predicts that for a sufficiently small spheroid (r ≪ rl ) that growth is
initially exponential, then inhibited by hypoxia and central anoxia. The critical radius
at which growth is no longer exponential, and where oxygen consumption limits the proliferating extent occurs at
rs = rl
r
1−
pm
.
po
(9)
These equations give rise to a classic sigmoid curve, where initial exponential growth is
followed by an approximately linear phase before growth begins to plateau. The model
predicts that the volume spheroids can obtain and the rate at which they grow is heavily influenced by the OCR and that high consumption rates result in decreased plateau
volumes relative to spheroids which consume oxygen at a lower rate. This is an interesting finding, as this models yields a sigmoidal shape akin to the observed experimental
curves mechanistically without any a priori assumption or forcing - some sample curves
are included in the supplementary material, in supplementary material S1: Figure 1.
These equations also predict that oxygen limited growth eventually plateaus, in line with
observations by Conger et al and others [3] - the radius at which this occurs can be
estimated by satisfying
ro3 = rp3 + rn3 .
(10)
The growth model predicts spheroid growth, using only the parameters of oxygen consumption rate a and average cellular doubling time td . In this work, we examine the
growth curves from spheroids from a number of different cell lines, and also outline a
clinical method for estimate OCR so that theoretical fits can be validated to clinical
data.
7
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Measurement of oxygen consumption and relationship with cellular mass
There is a degree of ambiguity in clinical terminology that is worth addressing here - the
quantity of oxygen gas consumed per unit time can be measured using extracellular flux
methods, and this is often referred to as OCR. However, this is a relative measurement of
oxygen consumed per cell per unit time. As a consequence, this could change markedly
between cell lines where average cell mass is different. In this work, we define OCR as
oxygen consumed per unit time per unit mass, as it facilitates cross comparison of OCR
between cell lines. The extracellular flux analysis for estimating oxygen consumption rate
per cell works by isolating an extremely small volume (typically less than 7µl) of medium
above a monolayer and measuring changes in the concentrations of dissolved oxygen for a
number of cells Nc . While the measured extracellular flux values, SH , allow quantitative
comparison of oxygen consumption per minute in a given cell line, more information is
required to allow comparison of OCR between cell lines. To circumvent this ambiguity,
we may establish a method of converting extracellular flux SH (in units of moles of oxygen per cell per minute) to OCR a (in S.I units of m3 kg−1 s−1 ) in order to facilitate
comparison between cell lines, factoring in cellular mass mc . This conversion is given by
a=
SH
0.032SH
=
60ρO2 Nc mc
1875ρO2 Nc mc
(11)
where ρO2 is the density of oxygen gas and Nc is the number of cells in the sample. The
factor of 0.032 kg arises in the equation from the fact that a mole of oxygen gas has a
mass of 32g. Typically at human body temperature of 37◦ C, ρO2 = 1.331 kg m−3 , and
as all cells in this work were maintained at this temperature we can employ this value.
Equation 11 indicates that a ∝ SH and a ∝ m1c , so information about cellular mass is
needed to completely describe the oxygen consumption characteristics of a cell line. If
cell volume can be estimated, then cell mass may be inferred by assuming that cells have
the density of water ρH2O . Mass is then given by
mc =
vc
ρH20
(12)
Oxygen consumption rate was measured using a Sea-horse extracellular flux analyzer
(Seahorse Bioscience, Massachusetts) and a range of values of SH for a range of cell
8
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numbers to ensure linearity for each cell lines. The raw Sea-horse data is included in the
supplementary material, S1: tables 1-6 inclusive.
Mass estimation technique
As cellular mass mc is related to the oxygen consumption rate by equation 5, it was
important to estimate this to facilitate comparison between OCR derived from theory
and experiment. To obtain approximate values for cell mass, we estimated the volume
of individual cells using 3D confocal microscopy. Cells (1 x 105 per well) from all lines
were seeded onto glass coverslips in a 6-well plate and incubated at 37 ◦ C for 24 hours; all
subsequent experiments were carried out at room temperature. Once cells had attached
to the coverslip, they were washed in serum-free culture medium and then incubated
for 5 min with PKH26 (2 µM; Sigma-Aldrich) prepared in Diluent C (Sigma-Aldrich) to
fluorescently label the cell membrane. An equal volume of 100% fetal bovine serum was
added for 1 min to stop the reaction. Cells were washed with regular culture medium,
fixed for 10 min with 4% paraformaldehyde, washed with phosphate-buffered saline, and
incubated with Hoechst 33342 (3.2 µM; Life Technologies) for 10 min to label nuclei. Coverslips were then washed in PBS and mounted using SlowFade r Gold Antifade Reagent
(Life Technologies) onto glass slides. Five fields of view from each cell line were acquired
using the 20x/0.87 M27 objective on a Zeiss LSM 710 microscope (Carl Zeiss AG). To
obtain cell volume estimates, each field of view was scanned using the z-stack feature,
with a 1 µm slice thickness, to obtain images for cell volume estimates as shown in Fig. 2.
To help counteract inherent blurring, unsharp masking was performed using a blurring
kernel and magnitude set manually to provide maximum edge definition. After the unsharp masking technique was applied, a MATLAB script was run which calculated the
area of each slice through a cell. These areas were summed and multiplied by the slice
thickness (1 µm) to estimate cell volume. Cell mass was estimated using the density
transform in equation 12. This method was validated by performing it on Hela cells, and
comparing the results to literature estimates of HeLa mass [27].
Cell culture and spheroid growth
Seven cell lines (ATCC) from different human cancers were cultured for in vitro experiments: the cervical carcinoma line HeLa (ATCC® CCL-2), colorectal lines HCT 116
(ATCC ® CL-247), LS 174T(ATCC ® CL-188), the breast cancer lines MDA-MB-468
(ATCC ® HTB-132) and MDA-MB-231 (ATCC ® HTB-26), the squamous cell carcinoma
line SCC-25 (ATCC CRL-1628), the glioblastoma cell line U-87 MG (ATCC HTB-14).
9
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Figure 2: Representative images (maximum intensity projections) of five cell
lines that were fluorescently labelled for estimating cell volume using 3D
confocal microscopy. Merged images show the simultaneous staining of the nucleus
(blue, Hoechst 33342, 3.2µM) and the cell membrane (green, PKH26, 2µM) in (A) HeLa
cells and in (B) the MDA-468, HCT116, Ls-174T, and SCC25 cells. Scale bars represent
10µm
All cell lines were grown in Dulbecco’s Modified Eagle Medium (Life Technologies) supplemented with 10% fetal bovine serum, with the exception of the SCC-25 line, which
was cultured in growth medium recommended by the supplier (American Type Culture
Collection). Cells were maintained at 37 ◦C in humidified incubator containing 5% CO2 .
Spheroids were generated as previously described [28] in literature. Briefly, 0.2 µl of
2.5 ×104 /ml cell suspension was added to each well of a 96- round-bottom-well ultra low
attachment plate (Corning Incorporated) in a high glucose medium (4.5g /L). For MDAMB-468 and SCC25 cells, Matrigel (BD Bioscience) was added at a final concentration of
5%. Plates were then centrifuged at 300xg for 10 min.. Centrifugation was carried out at
4◦ C if Matrigel was required for spheroid formation. Spheroids were maintained at 37◦ C
in a humidified incubator containing 5% CO2 . Pictures of spheroids were taken using
the EVOS XL Core Cell Imaging System (Life Technologies) and volumes were analysed
using the ImageJ (NIH) . Hela cells were not grown as spheroids, but were used to validate the mass estimation method outlined previously. The oxygen diffusion constant D
was taken to be close to that of water, approximately D = 2 × 10−9 m2 /s [14, 26, 29].
To minimise fluctuations in glucose concentration, we renewed spheroid medium every 48
hours by replacing half the volume of medium in each well with the equivalent volume of
fresh culture medium.
10
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Model validation
Growth curves were obtained for spheroids for each cell line - table 1 lists a summary
of all cell-lines. The initial volume of each spheroid line was measured and from this ro
calculated. The oxygen limit for mitotic arrest was taken from literature to be pm = 0.5
mmHg for spheroids in a glucose solution [30]. The best-fit between model and the experimental growth curves was then calculated. When possible, the OCR for each cell line
was then estimated using the experimental procedure outlined. This data was then used
to produce a theoretical growth curve which could be directly compared to the experimental data without degenerate fitting.
The model could also be used to estimate the effect of different clinical compounds of
OCR - to investigate this, 11 HCT 116 spheroids were grown, and 7 of these treated
with 50nM of gemacitabine. The remaining 4 functioned as experimental controls. These
were stained with the proliferation marker Ki-67 and the hypoxia marker EF5, and sectioned through the centre. The OCR was estimated used sectioning techniques [26] and
the resulting OCRs were compared for both groups using a two-tailed Welch’s correction
t-test.
Table 1: Summary of Spheroids used for growth curves
Cell Line
MDA-MD-231
U-87
SCC-25
HCT 116
LS 174T
MDA-MB-468
Spheroid Formation Method
Suspension in low-attachment plate
Suspension in low-attachment plate
Suspension with 5% Matrigel in low-attachment plate
Suspension in low-attachment plate
Suspension in low-attachment plate
Suspension with 5% Matrigel in low-attachment plate
11
Number of spheroids
18
18
18
24
30
18
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Results
0.1
Model fitting
Theoretically derived growth curves were fit to experimental data for a range of cells lines,
and from the best fit parameters used to estimate OCR a and average cellular doubling
time td . This analysis imposes a constraint condition on the data of
G = log2
Vmax
Vmin
,
(13)
where Vmax is the maximum spheroid volume at the end of the growth period and Vmin the
initial volume at t = 0. For model fitting purposes, the condition G ≥ 2 ensures that a
least two cell doubling times of td have transpired and avoids over-fitting. Such a consideration ruled out the use of certain cell lines such as T 47D (ductal carcinoma) as volume
increases were too small for analysis over the growth period. Growth curves were fitted
for three distinct cell lines; MDA-MB-231 Breast adenocarcinoma, U-87 Glioblastoma,
and SCC-25 squamous cell carcinoma. Fits were also obtained on previously published
data by Freyer [4, 17] for V-79 hamster fibroblast cells. This data was selected as it is
relatively long range ( 60 days) and plateau effects can be readily observed. Fig. 3 shows
the ideal theoretical fits for these curves which yields the greatest co-efficient of determination, indicating that the model fits the data extremely well. Error bars on the time axis
are ± 12 a day to capture uncertainty on exact time which growth curves were measured
on a daily basis. It is important to note however that doubling time td and diffusion limit
are degenerate parameters and there exist a considerable range of parameters which will
yield similar fits. This degeneracy is explored further in the discussion.
12
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10
x 10
10
(a) MDA MB 231 (Breast adenocarcinoma)
6
4.5
(b) U87 (Glioblastoma)
x 10
2
2
a = 32.89 mmHg / s, td = 2.85 days. R = 0.99
a = 41.04 mmHg / s, td = 1.21 days, R = 0.99
Experimental data
Experimental data
5
4
3.5
4
3
Volume (m )
3
Volume (m )
3
2.5
3
2
2
1.5
1
1
0.5
0
0
2
9
9
4
6
8
10
Time (Days)
12
14
16
0
18
10
(c) V 79 (Hamster lung fibroblast)
x 10
8
x 10
10
8
6
4
Time (Days)
2
0
(d) SCC25 (Squamous cell carcinoma)
a = 24.79 mmHg / s, td = 2.81, R2 = 0.99
2
a = 6.8 mmHg / s m, td = 2.8 days, R = 0.96
Experimental data
Experimental data
8
7
7
6
6
3
Volume (m )
3
Volume (m )
5
5
4
4
3
3
2
2
1
1
0
0
10
20
30
Time (Days)
40
50
0
0
60
2
4
6
10
8
Time (Days)
12
14
16
18
Figure 3: Theoretical best fits for (a) MDA-MB-231 (b) U-87 and (c) Hamster
V-79 spheroids. Data for the V-79 cells is from previously published investigations by
Freyer [4] and standard errors are not shown on this plot. (d) SCC-25. While best fits
are shown in this figure, there are several possible combinations of diffusion limit (rl )
and doubling time (td ) that produce similarly high co-efficients of determination so these
results may not be uniquely determined.
13
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Comparison of theoretical curves to experimental data
As curve-fitting suggests the model fits the data well, it is possible for some cell lines
to avoid potential degeneracy and directly contrast theoretical curves with experimental
data, provided OCR can be determined. In this case curve-fitting is not required and
model and data can be directly compared. Cell volume and hence mass estimates were
obtained for a number of cells in four distinct cell lines; HCT 116 (n = 36), LS 147T
(n = 36), MDA-MB-468 (n = 27) and SCC-25 (n = 22). For these cell lines, multiple
individual cells could be isolated and cell mass was estimated by the procedure outlined
in the methods section. This was combined with the extracellular flux measurements SH
to yield an estimate for the consumption rate a and the resultant OCR a. Best estimates
for oxygen consumption are shown in table 2. The diffusion constant was assumed to
be close to that of water so D = 2 × 10−9 m2 s−1 . The oxygen partial pressure in the
medium at the spheroid boundary was po = 100 mmHg. Results for these cell lines are
shown in figure 4, where model results are contrasted to experimental data. Results are
shown with their respective best fit doubling times, td . Error bars on the time axis are
± 21 a day to capture uncertainty on exact time which growth curves were measured on
a daily basis. The model data illustrated in figure 4 are independent of fitting, directly
contrasting the model with OCR taken from the experimental data in table 1.
14
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(a) HCT 116 (Colorectal carcinoma)
−10
8
x 10
x 10
a = 20.61 mmHg / s, t = 1.34 days, R2 = 0.9858
a = 33.97 mmHg / s, t = 2.16 days, R2 = 0.9677
a = 24.98 mmHg / s, t = 1.07 days, R2 = 0.9560
2
2
2
7
(b) LS 174T (Colorectal adenocarcinoma)
−9
1.2
a = 27.92 mmHg / s, t = 2.87 days, R2 = 0.9958
2
a = 21.87 mmHg / s, t2 = 3.69 days, R2 = 0.9967
a = 16.23 mmHg / s, t2 = 1.63 days, R2 = 0.9980
1
Experimental data
Experimental data
6
0.8
Volume *m3)
3
Volume (m )
5
4
0.6
3
0.4
2
0.2
1
0
0
2
4
8
10
Time (Days)
12
14
16
0
0
18
(c) MDA−MB−468 (Breast adenocarcinoma)
−9
1.2
6
x 10
2
3
4
Time (Days)
5
6
7
8
14
16
18
(d) SCC−25 (Squamous cell carcinoma)
−9
1
x 10
2
a = 11.21 mmHg / s , t = 4.05 days, R2 = 0.9327
a = 18.09 mmHg / s , t = 2.25 days, R = 0.9388
2
2
2
a = 22.61 mmHg / s , t = 1.91 days, R = 0.8930
0.9
2
1
1
a = 13.56 mmHg / s , t2 = 2.61 days, R2 = 0.9714
a = 15.84 mmHg / s , t = 3.67 days, R2 = 0.9677
2
a = 6.58 mmHg / s , t2 = 4.29 days, R2 = 0.8838
Experimental data
0.8
Experimental data
0.7
0.8
Volume (m3)
3
Volume (m )
0.6
0.6
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0
0
2
4
6
8
Time (Days)
10
0
0
12
2
4
6
8
10
Time (Days)
12
Figure 4: Plots of experimental data and model growth curves for (a) HCT
116 (b) LS 174T (c) MDA-MB-468 and (d) SCC-25 spheroids. In all plots
the growth curve due to mean experimentally estimated OCR a is denoted by a solid
blue line, with one standard deviation above average OCR marked by a dashed red line
and one standard deviation below average consumption marked with a dotted green line.
Best fit doubling times td and co-efficient of determination are shown for each value with
high goodness of fit obtained for each estimated consumption rate within the confidence
intervals of experimental data. The shaded area corresponds to range of ± 2 standard
deviations for OCR.
15
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Table 2: Experimentally measured cell mass / OCR
Cell Line
HCT 116
LS 174T
MDA-MB-468
SCC-25
Mass (ng)
2.34 ± 0.45
2.12 ± 0.39
3.22 ± 0.61
3.54 ± 1.06
Seahorse data (pM / cell)
5.37 ± 0.13 ×10−3
3.59 ± 0.10 ×10−3
4.79 ± 0.30 ×10−3
3.27 ± 0.37 ×10−3
a (m3 kg −1 s−1 )
9.21 ± 1.99 ×10−7
6.80 ± 1.45 ×10−7
5.97 ± 1.49 ×10−7
3.70 ± 1.53 ×10−7
a (mmHg / s)
27.92 ± 6.05
20.61 ± 4.38
18.08 ± 4.53
11.21 ± 4.63
Effects of clinical compounds on OCR
Using the model, the effects of gemcitabine on OCR could also be ascertained. This is
illustrated in figure 5, for untreated HCT-116 spheroids and HCT-116 spheroids treated
with 50nM of gemcitabine. Untreated HCT 116 spheroids were estimated to have an
average OCR of 27.43 mmHg/s, in high agreement with estimated consumption rate
experimentally derived in this work (mean value 27.92 mmHg / s )through the confocal
mass estimation method. A Welch’s correction two-tailed T-test was performed between
the two groups, with highly significant result of P < 0.01. These results are shown in
Fig. 5.
16
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(b)
(a)
(c)
12
OCR ( x 10−7 m3 kg−1 s−1)
10
8
6
4
OCR Untreated spheroids
OCR with 50nM Gemcitabine
Average untreated OCR
Averaged OCR with 50nM Gemcitabine
2
0
Figure 5: (a) A HCT-116 control spheroid stained for proliferating cells using Ki-67 (green) and for hypoxia using EF5 (red) (b) a HCT 116 spheroid
treated with 50 nM of gemcitabine showing markedly smaller hypoxic centre than untreated spheroid. (c) OCR estimated from stained cross-sections
by previously outlined method [26] for 4 control spheroids and 7 spheroids
treated with 50nM gemcitabine. Average OCR for treated spheroids is 6.18 ×10−7
m3 kg−1 s−1 (18.75 mmHg / s) versus 9.05 ×10−7 m3 kg−1 s−1 (27.43 mmHg / s) for untreated spheroids (P-Value < 0.01 using a two-tailed Welch’s correction t-test,α = 0.05).
This suggests a marked decrease in OCR for treated spheroids.
17
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Mass validation
To quantify the accuracy of the image analysis method outlined in the prior section,
HeLa cells were imaged with the confocal microscope and subsequently deblurred using
the unsharp masking technique and area detection algorithm. A selection of these cells
(n = 15) were then run through the area detection code in MATLAB so that volume, and
hence mass, could be estimated. This yielded an estimated mass of 2.95 ± 0.54 ng for the
HeLa cells analysed. This is in good agreement with mass estimates using a cantilever
method [27] (3.29 ± 1.14 ng).
Discussion
This work outlines a simple discrete model for spheroid and avascular tumor growth,
quantifying proliferating volume after mitosis with OCR and doubling time as the free
parameters. The simple model outlined in this work includes only a minimum of terms
for which parameters are either measurable or known. Despite its relative simplicity,
the model replicates the growth behaviour of spheroids well for a range of different cell
lines of different types, providing a mechanistic explanation for the observed sigmoidal
curves associated with spheroid growth. What is particulary worthy of note is the effect of consumption rate on the growth that this analysis suggests, with higher rates of
oxygen consumption resulting in a markedly lower plateau volumes. The relatively intuitive reason for this is that the rate of oxygen consumption directly influences the size
of the viable spheroid rim [26], as the distance oxygen diffuses is related to how rapidly
the respiring tissue consumes it. Consequently, increased oxygen consumption suggests
a greater anoxic centre and thinner viable rim, and a smaller volume of proliferating cells.
For the MDA-MB-231, U-87, SCC-25 and V-79 cell lines, theoretical fitting methods
were employed to find best OCR and doubling time parameters for a given growth curve.
The resultant fits in good agreement (0.96 ≤ R2 ≤ 0.99) for all cases. There is some
unavoidable uncertainty on these fits due to the fact that doubling time td and diffusion
limit rl (and by extension OCR) are degenerate parameters, as illustrated in Fig. 6. In
principle if the OCR can be estimated, then this degeneracy can be circumvented.
For four cell lines, it was possible to use the OCR determination method outlined to
obtain an estimate of OCR, circumventing degeneracy and testing the model further.
With a known OCR, model growth curves could be directly contrasted to experimental
data. This model validation was performed on growth curves for spheroids from a range
of cell-lines, namely the HCT 116, LS 174T, MDA-MB-468 and SCC-25 cell lines. OCR
18
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Figure 6: Best fit degeneracy for U-87 growth curve. While most values of rl / td
yield negative co-efficients of determination, there is a relatively narrow-band (shown in
color) that produces a good fit to observed data (R2 > 0.95). In this case, values of rl
between 160 - 215 µm (8.56 - 15.46 ×10−7 m3 kg−1 s−1 ) can yield good fits, with these
values yielding doubling times between 0.6 - 2.1 days. The range value is due to inherent
degeneracy between diffusion limit and doubling time
was experimentally estimated using extracellular flux analysis combined with mass estimates derived from confocal microscopy and these estimates were then used to produce
a growth curve, which was contrasted with measured spheroid growth curves for that cell
line. From this, the best-fit cellular doubling time could be estimated and the agreement
between experimental and theoretical curves quantified. For the cell lines analysed this
way, agreement was high with mean co-efficient of determinations ranging from 0.9327 to
0.9958, suggesting the model is robust and describes the data well. This analysis was also
attempted on the MDA-MB-231 and U-87 lines, but cell mass estimates of these lines
were confounded by inability to accurately resolve individual cells, due either to high
levels of mitotic cells (MDA-MB-231) or highly irregular cell shape (U-87). The SCC-25
line had considerable uncertainty in mass estimation (≥ 40%) which rendered its OCR
uncertain and different from that in the pure curve fitting section.
Derived values for OCR lend themselves to estimations of cellular doubling time. Literature reports of doubling time have wide ranges, even for single cell lines, suggesting
19
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this may be heavily influenced by the conditions under which the cells are grown or the
method used to estimate doubling time. For HCT 116, LS 174T, MDA-MB-468 and
SCC-25, recent literature estimates of doubling time are 1.25 days, 1.33 days, 2 days and
2.1 days respectively [31–33]. This is in very good agreement with the estimated value
for the LS 174T and MDA-MB-268 lines used in this work, but lower than estimated for
the HCT 116 and SCC-25 lines. Interestingly, when SCC-25 was theoretically fit, the
estimate for doubling time reduced to 2.85 days, closer to literature values than from the
experimental case. There is also the possibility that cells in a spheroid grow differently
to plated cells, and indeed there is some recent evidence for geometry specific metabolic
differences in specific situations [34]. In theory, this could be tested by growing very small
spheroids in their exponential phase of growth (ro << rc ) and estimating the doubling
time of the entire spheroid, however it is technically challenging to do so. It is also quite
likely that the doubling time of cells is influenced by their growth conditions, which may
render such comparisons void. In any case, the OCRs estimated from this method are in
the same range as typical literature estimates. It was also possible to estimate HTC-116
OCR using two methods - this analysis yielded results within ≈ 1.7% of one another.
There are several potentially confounding factors in this work; one of which is the minimum oxygen tension required for mitosis, pm , taken to be 0.5 mmHg for all cell lines in
this work [30]. It is possible however that hypoxic arrest limits differ between cell types.
Higher values of pm would mean a decreased proliferating volume Vp and consequently a
decreased maximum volume. Conversely, lower values would act to increase Vp . Whether
this varies between cell lines is an open question. The effects of glucose are not modelled
- this was because the spheroids were grown in a high glucose media (25mM or 4.5g/L)
which suggests spheroids should be well supplied, and that the effects of low-glucose could
be ignored without loss of generality. However, we can in principle use the model derived
in this work to estimate what effect this might have; for low glucose environments, the
literature estimate for minimal oxygen partial pressure for mitosis raises an order of magnitude to pm = 5 mmHg [30]. In this case, the proliferating volume Vp would be markedly
reduced. This situation is illustrated for a hypothetical spheroids in two different glucose
concentrations in the supplementary material (S1: Figure 2), where it can be seen that
even with an order of magnitude change in concentration, the effects are relatively small.
Another potentially confounding factor that may have considerable effect is the cellular density; in the results shown this was approximated to the density of water, but there
is some evidence this can vary with cell cycle, and can be between 4% and 9% higher than
the density of water [35]. If the higher estimates for density are used, this acts to increase
20
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the estimate of cellular mass and consequently reduces the estimated consumption rate as
outlined in equation 5. For the fits shown in figure 4 (a) - (c), this slightly improves the
co-efficient of determination. In the case of SCC-25, this has the opposite effect, perhaps
related to the suspect samples for this particular cell line. The fit data for higher density
estimates is not shown for brevity.
The model also makes an implicit assumption in the model is that the rate at which
necrotic material is removed from anoxic core is sufficiently large that proliferation into
the core region is not limited by space, and new cells are either pushed into available
space in the core or bump existing cells into that void where they die due to insufficient
oxygen. This inward migration of cells might be a consequence of several factors, but it
is likely that high oxygen pressure around the spheroid or simply surface tension would
act to confine the volume and push emerging cells into available central space. Were
this not the case, one would expect spheroid radius to increase linearly and volume exponentially without limit, which is not observed in practice, instead spheroids are seen
to progress from linear growth to a plateau phase [3, 4]. This is also suggested by the
model; initially the proliferating volume is greater than the anoxic core volume and so the
spheroid experiences net growth, with an increasing anoxic core volume as cells die due
to inadequate oxygen supply. Eventually, the anoxic volume is sufficiently large to absorb
any new cells without net increase; equation 10 suggests that a plateau state is achieved
p
when ro ≥ 3 rp3 + rn3 . Re-arrangement of this identity shows this is the same as writing
that plateau is achieved when the proliferating volume matches the anoxic core volume,
as might be expected mechanistically. The exact kinetics of cell lysate removal from the
core is not known, but imaging of sectioned spheroids appears to show an absence of cells
in the centre, implying this material is indeed removed. When this condition is reached
then even though the proliferating volume is constant and non-zero, the spheroids cannot
increase in size. This situation is illustrated in the supplementary material (S1: figure
3).
It is also clear from the identity derived in equation 5 that cellular mass is needed to
fully characterize OCR from extracellular flux analysis, and the range of masses estimated for cells in this work (2.12 - 3.54 ng) suggests cellular mass differs between cell
lines, and can introduce substantial errors into OCR estimation if neglected. An example
of this is seen in this work, where MDA-MB-468 cells produced higher values on the
seahorse analyzer (Seahorse Bioscience, Massachusetts) than LS 174T cells, but due to
the very small mass of the latter, the estimated OCR for LS 174T was in fact 14% higher
than estimated OCR for the MDA-MB-468 cells.
21
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The cell mass estimation in this work also has a number of potentially confounding
factors. If cells are undergoing mitosis or apoptosis, these will tend to shew the volume
estimates. To try and over-come this, both a membrane stain and nuclear stain were
used so obviously unsuitable cells could be excluded from analysis, and a number of cells
(22 ≤ n ≤ 36 ) were analyzed for each cell line. In other lines, cells were too close together
to resolve (U-87) or all mitotic (MDA-MB-231) rendering the confocal volume approach
of limited use. The volume reconstruction algorithm used here was relatively simple, and
re-constructed volume estimates based on one micron slices. This might introduce errors
for unevenly spaced cells. This could potentially be overcome with finer spacing, but this
would act to increase bleed-through error and increase processing time. However, the
majority of the derived mass estimates produced OCRs which matched the growth data
well and mass validation with HeLa cells was encouraging.
The use of the model to estimate OCR change due to clinical compounds is also of
interest. HCT-116 Spheroids treated with 50nM of Gemcitabine had a much lower OCR
(mean 18.75 mmHg/s) than control HCT-116 spheroids (mean 27.43 mmHg / s) with
OCR measured by the sectioning method [26]. Encouragingly, estimation of OCR for
untreated HCT-116 spheroids by the sectioning method was in very close agreement
(< 1.8%) to that determined in this work using the OCR technique outlined (mean 27.92
mmHg / s). The stark effect of Gemcitabine is also interesting; it is a known and potent
radio-sensitizer [36–38], although the exact mechanism of action is still untested. This
analysis might suggest that this drug reduces OCR, allowing greater oxygen diffusion and
decreasing hypoxic regions, though more analysis would be required to test this hypothesis further. The spheroids in this work were stained with the proliferation marker Ki-67,
which shows all cell cycle phases except G0; this yields a binary distinction between quiescent and non-quiescent cells. For more detailed investigation of cell cycle state, more
specific markers such as BrdU might be appropriate.
Spheroids in this work ranged in radius value from 182 µm to 617 µm, and OCR throughout growth was assumed to be constant for cell lines observed. This assumption is supported by analysis of DLD-1 colorectal spheroids which yielded an approximately constant oxygen consumption rate between 370 and 590 µm [26], though analysis of other
cell lines by different methods suggests that consumption rate can vary up to 50% in
some lines whilst changing minimally in others [39]. This assumption is supported by a
recent theoretical analysis [29] which examined hyperbolic Michaelis-Menten like oxygen
consumption forms and found minimal variation between these forms and the simpler as22
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not peer-reviewed) is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license.
sumption of constant consumption rate. For spheroids in this analysis, a constant OCR
assumption fits the data but if OCR variation is known for a given cell line it would be
readily incorporated into the model. Although many types of spheroid culture methods
exist, we used an ultra-low attachment format to grow spheroids for experimental simplicity and for volume measurements. The model presented here is relatively simple, and
could be readily extended to incorporate other effects if the underlying parameters can be
measured. There is also the possibility that oxygen dynamics differ between monolayers
and spheroids, which would serve to confound the derived OCR data presented in table
2. This is an avenue worthy of further investigation, and it might be worthwhile to section stained spheroids to estimation their OCR by previously outlined methods [26] and
contrast this with the OCR method outlined here. While there are numerous models in
the literature for avascular spheroid growth [3,20–24], this model is novel as it specifically
relates growth and growth limitation to oxygen status and availability.
Conclusions
The model presented in this work yields projected spheroid growth curves from knowledge
of oxygen consumption rate and cellular doubling time. Theoretical growth curves were
found to match experimental data well for a range of cell lines with different characteristics, yielding the classic sigmoidal shape expected mechanistically without any a priori
assumptions. This work also illustrates the importance of cellular mass is ascertaining
OCR, and outlines a method for estimating this. Finally, a model was applied to infer
the change in OCR due to clinical compounds, demonstrated with gemcitabine as a proof
of principle.
Supporting Information
S1 Figure 1
Growth curve arising from oxygen model The assumption of oxygen mediated
growth gives rise to the classic sigmoidal growth curve.
S1 Figure 2
Growth curves as a function of glucose availability Lower glucose levels give rise to
a modified growth curve as pm increases when glucose isn’t available. Despite pm changing
by up to a factor of 10 under such circumstances, the growth curves under conditions of
both ample glucose and glucose deficiency are quite similar.
23
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S1 Figure 3
Relative radii of spheroid sections with growth For a spheroid of initial radii 20
µm and a = 7 × 10−7 m3 kg−1 s−1 , the proliferating rim radius is the same as the spheroid
radius. After r = rc , the proliferating radius falls relative to the anoxic radius rn which
p
increases. At plateau, ro = 3 rp3 + rn3 and no further increase in volume is observed.
S1 Tables 1-6
Raw Seahorse oxygen consumption data for the range of cells used in this
work.
Acknowledgments
The authors gratefully extend their thanks to Prof. James Freyer for kindly allowing them
use his V79 hamster cell spheroid data, and thank the reviewers for their suggestions on
the kinetics of the necrotic core, and Dr. Daniel Warren of University of Oxford for his
useful feedback.
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